Abstract: For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove that the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is ``doubly ruled'' (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define ``doubly ruled parameterized surfaces'' in any Riemannian manifold, independent of Riemannian submersions. We initiate a study of the doubly ruled parameterized surfaces in compact Lie groups and in other symmetric spaces by establishing several rigidity theorems and constructing examples.